Simplify the following expression and state the condition under which the simplification is valid. You can assume that $p \neq 0$. $z = \dfrac{12p^2 + 4p}{7p} \div \dfrac{5(3p + 1)}{-4} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $z = \dfrac{12p^2 + 4p}{7p} \times \dfrac{-4}{5(3p + 1)} $ When multiplying fractions, we multiply the numerators and the denominators. $z = \dfrac{ (12p^2 + 4p) \times -4 } { 7p \times 5(3p + 1) } $ $ z = \dfrac {-4 \times 4p(3p + 1)} {7p \times 5(3p + 1)} $ $ z = \dfrac{-16p(3p + 1)}{35p(3p + 1)} $ We can cancel the $3p + 1$ so long as $3p + 1 \neq 0$ Therefore $p \neq -\dfrac{1}{3}$ $z = \dfrac{-16p \cancel{(3p + 1})}{35p \cancel{(3p + 1)}} = -\dfrac{16p}{35p} = -\dfrac{16}{35} $